Question

1. f(x) = 2x² - x + 1

g(x) = 4x + 3

Explanation:

Assuming we need to find the composition of functions (e.g., \( f(g(x)) \) or \( g(f(x)) \)), let's take \( f(g(x)) \) as an example.

Step1: Substitute \( g(x) \) into \( f(x) \)

We know \( f(x) = 2x^2 - x + 1 \) and \( g(x) = 4x + 3 \). So we substitute \( x = g(x) = 4x + 3 \) into \( f(x) \).
\( f(g(x)) = 2(4x + 3)^2 - (4x + 3) + 1 \)

Step2: Expand \( (4x + 3)^2 \)

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), where \( a = 4x \) and \( b = 3 \), we get \( (4x + 3)^2 = 16x^2 + 24x + 9 \).
So \( f(g(x)) = 2(16x^2 + 24x + 9) - (4x + 3) + 1 \)

Step3: Distribute the 2

\( 2(16x^2 + 24x + 9) = 32x^2 + 48x + 18 \)
Now the expression becomes \( 32x^2 + 48x + 18 - 4x - 3 + 1 \)

Step4: Combine like terms

Combine the \( x \) terms: \( 48x - 4x = 44x \)
Combine the constant terms: \( 18 - 3 + 1 = 16 \)
So the final expression is \( 32x^2 + 44x + 16 \)

Answer:

If finding \( f(g(x)) \), the result is \( \boldsymbol{32x^2 + 44x + 16} \) (if the intended operation was different, adjust the steps accordingly).